Consider a sealed-bid second price auction with three bidders. The seller has a value 0 for the object. Each bidder has a value drawn from the uniform distribution F(v) = v on [0, 1], i.e., the probability that his value is less than x is just x.

suppose the seller can also submit a bid r, which is his "reservation bid". If the highest bid is lower than r, then the object is not sold. If the highest bid is greater than r, then the highest bidder gets the object at a price equal to the larger of the other two bids, or r, whichever is greater. For an arbitrary r, what is the probability that:

i. Both the highest and second highest bids exceed r,

ii. Exactly one buyer bids more than r (be careful, you need to count the probability that it is bidder 1 or bidder 2 or bidder 3),

iii. All buyers bid less than r?